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\markboth{\hfil First-order differential equations \hfil EJDE--2001/40}
{EJDE--2001/40\hfil Leonid Berezansky \& Yury Domshlak \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 40, pp. 1--19. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
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%
First-order differential equations with several deviating arguments:
Sturmian comparison method in oscillation theory: I
%
\thanks{ {\em Mathematics Subject Classifications:} 34K11.
\hfil\break\indent
{\em Key words:} delay differential equations, several delays,
oscillation, non-oscillation, \hfil\break\indent
Sturmian comparison method.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted January 16, 2001. Published May 31, 2001.} }
\date{}
%
\author{ Leonid Berezansky \& Yury Domshlak }
\maketitle
\begin{abstract}
The Sturmian Comparison Method, elaborated previously by one of
the authors, is developed and applied to differential
equations with several deviating arguments.
For two delays, we obtain oscillation criteria that are explicit
and close to be necessary. We also present a comparison of our results
with those known in the literature.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[section]
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\catcode`@=11
\@addtoreset{equation}{section}
\catcode`@=12
\section{Introduction}
Differential equations with several deviating
(not necessarily delayed) arguments have been intensively
investigated for many years.
One of the problems studied for such equations is the
oscillation of solutions.
Other related questions which have been studied are the existence and
non-existence of positive solutions on a given finite interval or on
the semiaxis, and the existence of lower and upper bounds for the
length of sign-preserving intervals of a solution.
For autonomous equations where the coefficients and the
deviations of the arguments are constant
such problems were almost completely solved long time ago.
This means that some sufficient conditions which are rather close
to being necessary were obtained.
These results were based on the study of the characteristic quasi-polynomial,
which reduces many problems to the analysis and location of its roots.
Using this approach, each coefficient is taken into account
according to its contribution to the equation.
When a differential equation with deviating arguments is non-autonomous,
this technique cannot be applied and thus the problem
becomes much more complicated.
Nevertheless, oscillation theory of Delay Differential Equations
(DDE) has been developed; see, for example, \cite{e2,g2,l1}
and their references.
We can compare the complexity of the oscillatory properties of
DDE with only one retarded argument,
\begin{equation}
\label{1}
x'(t) +a(t)x[r(t)]=0,\quad t\geq t_0,
\end{equation}
with the complexity of the autonomous equation.
The equation with several deviations
\begin{equation}
\label{2}
x'(t) +\sum_{k=1}^n a_k(t)x[r_k(t)]=0,\quad t\geq t_0,
\end{equation}
has been much less studied. The present paper deals with several deviations,
but only in first-order DDEs. The study of the second order DDEs is
much more difficult.
The main shortcoming in a significant number of publications is that
the exact contribution of each coefficient
$a_k(t)$ and of each deviation $r_k(t)$
to the behavior of the solution of (\ref{2}) is not evaluated.
Unfortunately, most of the publications on the oscillation of equations
of type (\ref{2}) are based on methods that
cut off all the ``outstanding" parts of the equation.
The only exceptions being \cite{a1,e1,g1,h1,k1,l2,l3,n1,p1,y1},
we will compare their results with those obtained in the present paper.
Cutting off parts of the equation way lead to unsatisfactory results.
Although formally valid, they are rough and non-logically built.
In fact, (\ref{2}) is roughly reduced to (\ref{1}) and only afterwards is
investigated.
Moreover, if the coefficients $a_k(t)$ are of different sign, some of the
$r_k(t)$'s are delayed and some of them are advanced, such equations have
hardly been studied.
However, the situation with the relevant results in this field
and the possibilities of its further development is not so pessimistic.
The second author of the present paper developed the basis of
a rather efficient method for the investigation of the
oscillatory properties of DDEs.
This method is called the Sturmian Comparison Method (SCM)
\cite{d1,d2,d3,d4} by whcih some new results on the oscillation of (\ref{1})
and (\ref{2}) were obtained. In particular the results for
(\ref{2}) have not been improved or generalized untill now.
Moreover, these results catch the contribution of each $a_k(t)$ and $r_k(t)$,
can treat the case of a mixed deviating equation, and also provide an
estimate for the distance between two consecutive zeros of a solution.
Unfortunately, Monograph \cite{d1} was published in Russian
and has not been accessible for many scientists
working in this field.
That is why its results were almost unknown and did not influence
the publications in this area,
some of which are particular cases of the results obtained in
\cite{d1,d2}, both from the point of view of generality of the equations
and sharpness of the statements.
Discussions in recent years brought us to the conclusion that
SCM for DDEs with several deviations should be systematically presented,
including both new and known results, with an up-to-date review of the
literature.
We plan to publish several related papers on this topic.
The present paper contains the general theory of
SCM for (\ref{2}) (non-retarded arguments only).
Further, we apply this method to (\ref{2}) with nonnegative coefficients
$a_k(t)\geq 0$ and delays $r_k(t)\leq t$.
Later, we plan to consider (\ref{2}) with mixed deviating arguments and
with coefficients of different sign.
The results on the oscillatory properties of (\ref{1}) with only
one deviation argument are not in the framework of the present discussion.
Let us only mention that in \cite{d3,d4} the SCM is applied to this class
of equations.
\section{Sturmian comparison theorem}
Suppose $-\infty\alpha,\quad \beta>r_k(\alpha),\quad k=1,\dots,n
$$
and there exists a solution $y(t)$ of (\ref{7}) such that
\begin{equation}
\label{9}
y(\alpha)=y(\beta),\quad y(t)>0,\quad t\in (\alpha,\beta),\quad y(t)\leq 0,
\quad t\in E.
\end{equation}
The definition of RHC for (\ref{6}) and for the corresponding equation
is similar.
\paragraph{Definition} % def.2
(see \cite{d1})\\
a) A solution $x(t)$, $t_0\leq tT$. Otherwise, it is called
{\em quickly oscillatory}.
The following principal identity will be used
for obtaining the main results of this paper.
\begin{lemma}
For every function $x(t)$, $t\in (\alpha,\beta)$, with
continuous derivative and
for every function $y(t)$, $t\in [\alpha,\beta]\cup E$, with
continuous derivative,
we have
\begin{eqnarray}
\int_\alpha^\beta y(t)(lx)(t)\,dt
&=&\int_\alpha^\beta x(t)(\tilde{l}y)(t)\,dt +[x(\beta)y(\beta)
-x(\alpha)y(\alpha)] \nonumber \\
&&+\sum_{k=1}^n \Big\{\int_{(\alpha,\beta)\setminus i_k}[a_k(t)
-\tilde{a}_k(t)]x[r_k(t)]y(t)\,dt \label{11}\\
&&+\int_{i_k} a_k(t)x[r_k(t)]y(t)\,dt
-\int_{e_k} \tilde{a}_k(t)x[r_k(t)]y(t)\,dt\Big\}. \nonumber
\end{eqnarray}
\end{lemma}
The statement of this lemma is to check prove by direct calculations
using integration by parts.
The following statement is a direct and exact analogue
of the classical Sturmian Comparison Theorem
for the second-order ordinary differential equation
$ x''(t)+a(t)x(t)=0$.
\begin{theorem} \label{thm1}
Let $(\alpha,\beta)$ is RHC for (\ref{7}) ,
\begin{equation}
\label{12}
\tilde{a}_k(t)\geq 0,\quad t\in e_k,\quad k=1,\dots,n,
\end{equation}
\begin{equation}
\label{13}
a_k(t)\geq \left\{ \begin{array}{ll} 0,& t\in i_k,\\
\tilde{a}_k(t),& t\in (\alpha,\beta)\setminus i_k,
\end{array}\right.k=1,\dots,n.
\end{equation}
and at least one of (\ref{12})-(\ref{13}) is strict on some subinterval of $(\alpha,\beta)$.
Then (\ref{6}) has no positive solutions on $(\alpha,\beta)\cup \tilde{E}$.
\end{theorem}
\paragraph{Proof} The proof is based on the principal identity (\ref{11}).
Let (\ref{9}) hold for a solution $y$ of (\ref{7}) and let $x(t),\quad t\in
(\alpha,\beta)\cup E$
be a positive solution of (\ref{6}).
Then the left-hand side of (\ref{11}) is non-positive.
On the other hand (\ref{12}), (\ref{13}), (\ref{9}) imply that all
terms on the right hand-side
of (\ref{11}) are nonnegative and at least one term is positive.
We have a contradiction, and the statement is proved.
\begin{corollary} \label{coro1.1}
Suppose (\ref{7}) has a regular oscillatory solution and
$$
a_k(t)\geq \tilde{a}_k(t)\geq 0,\quad k=1,\dots,n,\quad t\geq t_0.
$$
Then (\ref{6}) has no positive solution on $(t_0,\infty)$.
\end{corollary}
\paragraph{Remark} % 1
Note the important fact that Theorem \ref{thm1} is concerned with the behavior
of solutions of DDE and DDI on a finite interval and not on a semiaxis.
Therefore one can obtain from Theorem \ref{thm1} not only explicit conditions of
oscillation, but also estimates of the length of the
sign-preserving intervals of the solutions.
\section{Construction of the ``testing equations"} %section 3
>From this section, we consider (\ref{2}) with two delays only.
The reason is the following: all the possible complications appear always
already for $n=2$,
while the computations are technically easier in this case.
The essence of the SCM is the following.
One needs to construct a family of ``testing"
(\ref{7}) for which a given interval $(\alpha,\beta)$ is RHC and
condition (\ref{13}) holds. This family must be as rich as possible.
In the present paper we are going to do it for the case
of two retarded arguments.
Further, we can use Theorem \ref{thm1} for (\ref{6})
to obtain explicit oscillation criteria.
Consider (\ref{6}), the corresponding differential equation and
(\ref{7}) for the case $n=2$ and $r_i(t)\leq t$, $i=1,2$:
\begin{equation}
\label{14}
(lx)(t):= x'(t)+a_1(t)x[r_1(t)]+a_2(t)x[r_2(t)]\leq 0,
\quad t\in (\alpha,\beta),
\end{equation}
\begin{equation}
\label{15}
(lx)(t):= x'(t)+a_1(t)x[r_1(t)]+a_2(t)x[r_2(t)]=0,\quad t\in (\alpha,\beta),
\end{equation}
\begin{equation}
\label{16}
(\tilde{l}y)(t):= -y'(t)+\tilde{a}_1[q_1(t)]q_1'(t)y[q_1(t)]+
\tilde{a}_2[q_2(t)]q_2'(t)y[q_2(t)]\geq 0,\quad t\in(\alpha,\beta).
\end{equation}
Denote
$$
\rho(t):=\min\{r_1(t);r_2(t)\},\; Q(t):= \max\{q_1(t);q_2(t)\},\;
R(t):=\max\{r_1(t);r_2(t)\}.
$$
It is obvious that functions $\rho(t)$ and $Q(t)$ are the inverse of each other.
Besides, in this particular case ($n=2$):
\begin{equation}
\label{17}
\begin{array}{llll}
e_j=[\beta,q_j(\beta)],&E=[\beta,Q(\beta)],
&i_j=[\alpha,q_k(\alpha)],&I=[\alpha,Q(\alpha)],\\[3pt]
\tilde{e}_k=[r_k(\alpha),\alpha],&\tilde{E}=[\rho(\alpha),\alpha],&
\tilde{i}_k=[r_k(\beta),\beta],&\tilde{I}=[\rho(\beta),\beta].
\end{array}
\end{equation}
\begin{lemma} \label{lem.2}
Let $\rho(\beta)>\alpha$ and let $\varphi(t)$, $k(t)$ be continuous
functions defined on $(\rho(\alpha),Q(\beta))$
satisfying the following conditions:
\begin{eqnarray}
\label{18}
&0\leq \int_\alpha^t \varphi(s)ds \alpha$
yield that $(\alpha,\beta)$ is
RHC for (\ref{16}).
From Theorem \ref{thm1} and Lemma \ref{lem.2} we obtain the following
the following theorem.
\begin{theorem} \label{thm2}
Assume Conditions (\ref{18})-(\ref{20}) hold , $\rho(\beta)>\alpha$,
and
\begin{equation}
\label{23}
a_i(t)\geq \left\{\begin{array}{ll}
0,&t\in (\alpha,q_i(\alpha)),\\[2pt]
\tilde{a}_i(t),&t\in (q_i(\alpha),\beta),
\end{array}\right. \quad i=1,2,
\end{equation}
where $\tilde{a}_i(t)$ are given by (\ref{21})
and at least one of the inequalities (\ref{23}) is strict
on some sub-interval. Then (\ref{14})
has no positive solution on $(\rho(\alpha), \beta)$.
\end{theorem}
\begin{corollary} \label{cor2.1}
Suppose the conditions of Theorem \ref{thm2} hold for a sequence of intervals
$(\alpha_k,\beta_k)$, $\alpha_k\to\infty$.
Then all solutions of (\ref{15})
are oscillatory.
\end{corollary}
\paragraph{Remark} % rem.2
No restrictions are imposed on the coefficients
$a_i(t)$ of (\ref{15}) outside the set
$\cup^{\infty}(\alpha_k,\beta_k)$ in Corollary \ref{cor2.1}.
\begin{corollary} \label{cor2.2}
Let $$ R(t)\to\infty,\quad \varphi(t)\geq 0,\quad
\int_{t_0}^{\infty}\varphi(s)ds =\infty,$$ and
conditions (\ref{19}), (\ref{20}), (\ref{23}) hold on $[t_0, \infty)$.
Then all the solutions of (\ref{15}) are oscillatory.
\end{corollary}
The following statement gives a clear proof of the well-known
fundamental oscillation criterion for the autonomous equation
\begin{equation}
\label{24}
x'(t)+a_1x(t-\tau_1)+a_2x(t-\tau_2)=0,\quad \tau_1\neq \tau_2,
\quad a_i>0, \quad \tau_i>0,\quad i=1,2.
\end{equation}
On the other hand, this statement demonstrates the sharpness of
Theorem \ref{thm2},
which in particular case (of autonomous equations)
allows to obtain necessary and sufficient conditions for the oscillation of all
solutions of (\ref{15}).
\begin{corollary} \label{cor2.3}
Suppose that for the characteristic quasi-polynomial of (\ref{24})
$$
F(\lambda):= \lambda+a_1\exp\{-\lambda \tau_1\}+a_2\exp\{-\lambda \tau_2\}
$$
the following condition holds:
\begin{equation}
\label{25}
F(\lambda)>0,\quad \forall \lambda \in (-\infty,\infty).
\end{equation}
Then all the solutions of (\ref{24}) are oscillatory.
\end{corollary}
It is obvious that (\ref{25}) is also necessary for
the oscillation of (\ref{24}) as well.
\paragraph{Proof.} It is clear that $F''(\lambda)>0$ for every $\lambda$.
Thus the equation
\begin{equation}
\label{26}
F'(\lambda) :=1-\tau_1 a_1e^{-\lambda\tau_1}-\tau_2 a_2e^{-\lambda\tau_2}=0
\end{equation}
has a unique root $\lambda_0$ and
$\inf_{-\infty0.
\end{equation}
The equality
$\tau_1 a_1e^{-\lambda_0\tau_1}+\tau_2 a_2e^{-\lambda_0\tau_2}=1$
implies the equivalence
\begin{eqnarray}
\lefteqn{F(\lambda_0)>0}\nonumber\\
&\Longleftrightarrow&
(\tau_1+\tau_2)\left(\lambda_0+a_1e^{-\lambda_0\tau_1}
+a_2e^{-\lambda_0\tau_2}\right)>0 \nonumber\\
&\Longleftrightarrow& 1+ \tau_2a_1e^{-\lambda_0\tau_1}+\tau_1a_2e^{-\lambda_0\tau_2}+
\lambda_0(\tau_1+\tau_2)>0 \nonumber\\
&\Longleftrightarrow &\frac{2}{\tau_1+\tau_2} + \frac{\tau_2-\tau_1}{\tau_1+\tau_2}
\left(a_1e^{-\lambda_0\tau_1}-a_2e^{-\lambda_0\tau_2}\right)>-\lambda_0
\label{28}\\
&\Longleftrightarrow& a_i\exp\big\{\tau_i\big[\frac{2}{\tau_1+\tau_2}+
\frac{\tau_2-\tau_1}{\tau_1+\tau_2}
\left(a_1e^{-\lambda_0\tau_1}-a_2e^{-\lambda_0\tau_2}\right)\big]\big\}
>a_ie^{-\lambda_0\tau_i},\nonumber\\
&& i=1,2\,. \nonumber
\end{eqnarray}
In Theorem \ref{thm2}, put
$$
\varphi(t):=\frac{\nu}{\tau_1+\tau_2},\quad k(t):=a_1e^{-\lambda_0\tau_1}-a_2e^{-\lambda_0\tau_2},
$$
where $\nu\in (0, \frac{\pi}{2}|\tau_2-\tau_1|)$ is a sufficiently small number
which will be chosen below.
Let $\alpha$ and $\beta$ be such that
$\beta-\alpha=\pi(\tau_1+\tau_2)/\nu$.
Then (\ref{18})-(\ref{19}) hold and
the inequalities
$$\displaylines{
k\tau_1=\tau_1 a_1e^{-\lambda_0\tau_1}-\tau_1 a_2e^{-\lambda_0\tau_2}
=1-\tau_2 a_2e^{-\lambda_0\tau_2}-\tau_1
a_2e^{-\lambda_0\tau_2}<1,\cr
k\tau_2=\tau_2 a_1e^{-\lambda_0\tau_1}-\tau_2 a_2e^{-\lambda_0\tau_2}
=\tau_2 a_1e^{-\lambda_0\tau_1}-(1-\tau_1 a_1e^{-\lambda_0\tau_1})>-1,
}$$
imply that $-\frac{1}{\tau_2}& \Big\{a_i e^{-\lambda_0\tau_i}+\frac{(-1)^i}{\nu}\Big[\frac{\nu \tau_j}{\tau_1+\tau_2}
-\sin \frac{\nu \tau_j}{\tau_1+\tau_2}\Big]
\left(a_1e^{-\lambda_0\tau_1}-a_2e^{-\lambda_0\tau_2}\right)\Big\}
\frac{\nu}{\sin\nu}, \nonumber
\end{eqnarray}
where $i=1,2$, $i\neq j$.
It is clear that (\ref{28}) is the limit form of (\ref{29})
as $\nu\to 0$.
Hence (\ref{29}) holds for sufficiently small $\nu$.
Theorem \ref{thm2} implies now that all solutions of (\ref{24}) are oscillatory.
\paragraph{Remark} % rm 3
Actually, (\ref{29}) implies something more:
each solution of (\ref{24}) has at least one change of sign
on every interval of length greater than
$(\pi(\tau_1+\tau_2)/\nu) +\max\{\tau_1,\tau_2\}$.
From Theorem \ref{thm2}, we will obtain an explicit condition for the oscillation
not only in pointwise terms, but in {\em the integral average terms} as well.
To avoid unwieldy formulations, we will not deal with the
estimations of the lengths of sign-preserving
intervals of the solutions (see the remark at the end of Section 2).
We will be confined to obtain an explicit condition for the
oscillation by Corollary \ref{cor2.2} only.
Readers interested in the first problem
are referred to \cite[Corollary 4.3.5, p.100]{d1}.
\begin{theorem} \label{thm3}
Let $\rho(t)\to\infty$ and suppose there exist functions $b_j(t)$,
$j=1,2$ such that
\begin{equation} \label{30}
a_j(t)\geq b_j(t)\geq 0, \quad j=1,2,\quad t\geq t_0;
\end{equation}
the limits
\begin{equation} \label{31}
B_{ij}:=\lim_{t\to\infty}\int_{\rho_i(t)}^t b_j(s)ds,\quad i=1,2,
\end{equation}
are finite with
\begin{equation} \label{32}
B_{11}+B_{22}>0;
\end{equation}
the system
\begin{equation}
\label{33}
\begin{array}{c}
(B_{11}B_{22}-B_{12}B_{21})x_1x_2-B_{11}x_1-B_{22}x_2+1=0\\
\ln x_1-B_{11}x_1-B_{12}x_2<0\\
\ln x_2-B_{21}x_1-B_{22}x_2<0
\end{array}
\end{equation}
has a positive solution $\{x_1;x_2\}$.
Then all solutions of (\ref{15}) are oscillatory.
\end{theorem}
\paragraph{Proof.} In view of (\ref{33}), the system
\begin{equation}
\label{34}
\begin{array}{c}
(1-x_1B_{11})\alpha_1-x_2B_{12}\alpha_2=0\\
-x_1B_{21}\alpha_1+(1-x_2B_{22})\alpha_2=0\\
\alpha_1+\alpha_2=1
\end{array}
\end{equation}
has a solution $\{\alpha_1; \alpha_2\}$, $\alpha_j>0$, $j=1,2$
(we omit the explanation).
For $i=1,2$, Denote
\begin{eqnarray}
P_i(\nu)&:=&\frac{\nu x_i}{\sin\nu \cos\nu(\alpha_2-\alpha_1)}
\label{35} \\
&&\times\exp\Big\{\left[-2\nu\alpha_i \cos\nu-(x_1B_{i1}-x_2B_{i2})\sin\nu(\alpha_2-\alpha_1)\right]
\frac{1}{\sin\nu}\Big\} \nonumber
\end{eqnarray}
From (\ref{33}) and (\ref{34}) it follows
$$
P_i(0)=\lim_{\nu\to 0} P_i(\nu) =
x_i\exp(-x_1B_{i1}-x_2B_{i2})<1,\quad i=1,2.
$$
Thus there exists $\nu_0>0$ such that for $\nu\in (0,\nu_0)$ the
inequalities $P_i(\nu)<1$, $i=1,2$ hold.
In Corollary \ref{cor2.2}, set
\begin{equation}
\label{36}
\varphi(t):= 2\nu\alpha_1 x_1 q_1'(t)b_1[ q_1(t)]+
2\nu\alpha_2 x_2 q_2'(t)b_2[ q_2(t)],
\end{equation}
\begin{equation}
\label{37}
k(t):= x_1q_1'(t)b_1[ q_1(t)]- x_2 q_2'(t)b_2[ q_2(t)].
\end{equation}
From (\ref{32}) it follows
\begin{equation}
\label{38}
\int_{t_0}^{\infty} \varphi(s)ds =\infty,
\end{equation}
\begin{equation}
\label{39}
\lim_{t\to\infty} \int_t^{q_i(t)} \varphi(s)ds =
2\nu\alpha_1 x_1 B_{i1}+2\nu\alpha_2 x_2 B_{i2}=2\nu\alpha_i,
\end{equation}
\begin{equation}
\label{40}
\lim_{t\to\infty} \int_t^{q_i(t)}k(s)ds= x_1 B_{i1}-x_2 B_{i2}.
\end{equation}
Besides, (\ref{20}) holds for $ t>T$, where $T$ is sufficiently large.
Indeed,
\begin{eqnarray*}
\lefteqn{\frac{k(t)}{\varphi(t)}\sin \int_t^{q_1(t)}\varphi(s)ds }\\
&=&\frac{x_1q_1'(t)b_1[ q_1(t)]- x_2 q_2'(t)b_2[ q_2(t)]}
{2\nu\left(\alpha_1 x_1 q_1'(t)b_1[ q_1(t)]+
\alpha_2 x_2 q_2'(t)b_2[ q_2(t)]\right)}
\sin \int_t^{q_1(t)}\varphi(s)ds \\
&\leq& \frac{1}{2\nu\alpha_1}
\sin \int_t^{q_1(t)}\varphi(s)ds
\end{eqnarray*}
and
$$
\lim_{t\to\infty} \int_t^{q_1(t)} \varphi(s)ds =\sin 2\nu\alpha_1.
$$
Since $\frac{\sin 2\nu\alpha_1}{2\nu\alpha_1}<1,$
$k(t)\sin\int_t^{q_1(t)}\varphi(s)ds\leq \varphi(t)$ for $t>T$.
Similarly, for $t>T$,
$k(t)\sin\int_t^{q_2(t)}\varphi(s)ds\geq -\varphi(t)$.
This implies that (\ref{20}) holds for $t>T$.
For $ i=1,2$, Denote
$$\displaylines{
C_i(t,\nu):= \frac{\nu x_i}{\sin \int_{t}^{q_1(t)} \varphi(s)ds+
\sin\int_{t}^{q_2(t)} \varphi(s)ds)} \cr
\textstyle
\times\exp\Big[-\int_{t}^{q_i(t)}\frac{\varphi(\xi)\cos\left(\frac{1}{2}\int_{\xi}^{q_1(\xi)}
\varphi(s)ds+
\frac{1}{2}\int_{\xi}^{q_2(\xi)} \varphi(s)ds\right)+k(\xi)
\sin (\frac{1}{2}\int_{q_1(\xi)}^{q_2(\xi)} \varphi(s)ds)}
{\sin \left(\frac{1}{2}\int_{\xi}^{q_1(\xi)} \varphi(s)ds+
\frac{1}{2}\int_{\xi}^{q_2(\xi)} \varphi(s)ds\right)}d\xi\Big].
}$$
It is easy to check that
\begin{equation}
\label{41}
\lim_{t\to\infty} C_i(t,\nu)=P_i(\nu).
\end{equation}
From $P_i(\nu)<1$, it follows
\begin{equation}
\label{42}
C_i(t,\nu)\leq 1,\quad t>T,\quad i=1,2.
\end{equation}
Condition (\ref{27}) implies
\begin{equation}
\label{43}
q_i'(t)a_i[q_i(t)]\geq q_i'(t)b_i[q_i(t)]C_i(t,\nu),\quad i=1,2.
\end{equation}
Equalities (\ref{33}) and (\ref{34}) imply
$$\displaylines{
\varphi(t)+2k(t)\nu\alpha_2=2\nu x_1q_1'(t)b_1[q_1(t)], \cr
\varphi(t)-2k(t)\nu\alpha_1=2\nu x_2q_2'(t)b_2[q_2(t)].
}$$
Hence (\ref{20}) holds for $t>T$.
All conditions of Corollary \ref{cor2.2} hold and the statement
is proved.
\begin{corollary} \label{cor2.4}
Suppose condition (\ref{27}) holds and the limit
\begin{equation}
\label{44}
\lim_{t\to\infty} \int_{\rho(t)}^t B(s)ds >\frac{1}{e}
\end{equation}
exists, where
$$
B(t):=\left\{\begin{array}{ll}
b_1(t),& if\ q_1(t)\geq q_2(t)\\ b_2(t),& if\ q_1(t)< q_2(t).
\end{array}\right.
$$
Then all solutions of (\ref{15}) are oscillatory.
\end{corollary}
\paragraph{Proof} The proof is similar to the one of Theorem \ref{thm2}.
We denote here
$\varphi(t):= \nu Q^{'}(t)B[Q(t)]$ and
$$
k(t):= \left\{\begin{array}{ll}
\frac{\varphi(t)}{\sin\int_t^{q_1(t)}\varphi(s)ds},& if\ q_1(t)\geq q_2(t),\\
-\frac{\varphi(t)}{\sin\int_t^{q_2(t)}\varphi(s)ds},& if\ q_1(t)< q_2(t).
\end{array}\right.
$$
\paragraph{Remark} %rm.4
In most cases for obtaining explicit oscillation conditions
one uses an integral $\int_{R(t)}^t$ but not $\int_{\rho(t)}^t$ (see,
for example,
the condition
\begin{equation}
\label{45}
\lim_{t\to\infty}\sup \int_{R(t)}^t \left(a_1(s)+a_2(s)\right) ds >1
\end{equation}
from \cite{h1}). It is not optimal, because
conditions like (\ref{45}) disregard all retarded arguments except
the $t-R(t)$, that is, the nearest to $t$.
But actually the oscillatory properties of the equation are determined
by the farthest delay and not by the nearest one.
\paragraph{Example 3.1} Consider the equation
\begin{equation}
\label{46}
x'(t)+a_1x(t-1)+a_2x[r(t)]=0,\quad t\geq 0,
\end{equation}
where $a_1, a_2>0$, $r(t)\leq t$, $\lim_{t\to\infty}
[t-r(t)]=0$, $r{'}(t)>0$.
In Theorem \ref{thm3}, put
$b_1(t):=a_1(t)=a_1$, $b_2(t):=a_2(t)=a_2$. Then
$$
\begin{array}{ll}
B_{11}=\lim_{t\to\infty}\int_{t-1}^t a_1ds=a_1,&
B_{12}=\lim_{t\to\infty}\int_{t-1}^t a_2ds=a_2,\\[3pt]
B_{21}=\lim_{t\to\infty}\int_{r(t)}^t a_1ds=0,&
B_{22}=\lim_{t\to\infty}\int_{r(t)}^t a_2ds=0.
\end{array}
$$
Hence system (\ref{33}) turns into
$$
\begin{array}{c}
-a_1x_1+1=0\\
\ln x_1-a_1x_1-a_2x_2<0\\
\ln x_2<0
\end{array}
$$
which is equivalent to the system
$$\begin{array}{c}
x_1=\frac{1}{a_1}\\
-\frac{1}{a_2}\ln(ea_1)\frac{1}{e}.
\end{equation}
Theorem \ref{thm3} implies that Condition (\ref{47})
is sufficient for the oscillation of all solutions of (\ref{46}).
This result confirms the intuitive conjecture that the asymptotic
behavior of (\ref{46}) is
close to the one of the equation
\begin{equation}
\label{48}
y'(t)+a_1y(t-1)+a_2y(t)=0.
\end{equation}
But it is well known that
Condition (\ref{47}) is necessary and sufficient for the
oscillation of all solutions of (\ref{48}).
Note that it is impossible to obtain this result
by Corollary \ref{cor2.4}.
\paragraph{Example 3.2}
Consider the equation
\begin{equation}
\label{49}
x'(t)+\frac{a_1}{t}x(\frac{t}{\mu})+\frac{a_2}{t}x(t-\tau)=0,\quad t\geq t_0>0,
\end{equation}
where $\mu>1$, $\tau >0$, $a_1,a_2>0$.
In Theorem \ref{thm3}, put
$$b_1(t):=a_1(t)=\frac{a_1}{t}, \quad b_2(t):=a_2(t)=\frac{a_2}{t}.
$$
Then $B_{11}=a_1\ln\mu$, $B_{12}=a_2\ln\mu$, $B_{21}=B_{22}=0$.
System (\ref{33}) turns into
$$\begin{array}{c}
-a_1x_1\ln\mu +1=0\\
\ln x_1-a_1x_1\ln\mu -a_2x_2\ln\mu <0\\
\ln x_2<0
\end{array}
$$
which is equivalent to
$$\begin{array}{c}
x_1=\frac{1}{a_1\ln\mu}\\
-\ln[a_1\ln\mu]-1\frac{1}{e\ln\mu}
\end{equation}
is sufficient for the oscillation of all solutions of (\ref{49}).
Note that (\ref{50}) does not depend on $\tau$.
\paragraph{Example 3.3}
Consider the equation
\begin{equation}
\label{51}
x'(t)+\frac{a_1}{t}x(\frac{t}{\mu})+\frac{a_2}{t^\beta}x(t-\tau)=0,\quad
t\geq t_0>0,
\end{equation}
where $a_1,a_2,\tau >0$, $\mu>1$, $0\leq\beta<1$.
In Theorem \ref{thm3}, let
$$
b_1(t):=a_1(t)=\frac{a_1}{t},\quad b_2(t):=\frac{A}{t}\leq a_2(t)=\frac{a_2}{t^\beta}, t>t_0,
$$
where $A$ is an arbitrarily large and positive constant. Then
$ B_{11}=a_1\ln\mu$, $B_{12}=A\ln\mu$, $B_{21}=B_{22}=0$.
One can repeat now all calculations in Example 3.2. Then (\ref{50}) is
$a_1\mu^{A}>\frac{1}{e\ln\mu}$ which holds for every $a_1>0$ for $A$
sufficiently large. Hence if $a_1>0$, $a_2>0$
then all the solutions of (\ref{51}) are oscillatory.
This result is rather unexpected.
Actually, for $\beta =0$ each one of the conditions
$ a_1>\frac{1}{e\ln\mu}$ and $a_2>\frac{1}{e}$
is necessary and sufficient for the oscillation of all solutions for the
``shortened" equations
$$
y'(t)+\frac{a_2}{t}y(\frac{t}{\mu})=0
$$
and
$$
z'(t)+a_2z(t-1)=0,
$$
respectively. For $\beta >0$ the ``shortened" equation
$$
z'(t)+\frac{a_2}{t^\beta}z(t-1)=0
$$
has a non-oscillatory solution for every $a_2>0$.
\paragraph{Example 3.4} Consider the equation
\begin{equation}
\label{52}
x'(t)+\frac{a_1}{t\ln t}x(t^{\alpha})+\frac{a_2}{t^\beta\ln t}x(t-\tau)=0,
\quad t\geq t_0>1,
\end{equation}
where $a_1,a_2,\tau >0$, $1>\alpha >0$, $1\geq\beta\geq 0$.
If $\beta=1$ and $a_1\alpha^{-a_2} >\frac{1}{e\ln\frac{1}{\alpha}}$
then all solutions of (\ref{52}) are oscillatory.
If $0\leq \beta <1$ then for every $a_1>0$, $a_2>0$
all solutions of (\ref{52}) are oscillatory.
The proofs of these two statements are similar.
\section{Non-oscillation criteria}
The aim of this section is to show that the oscillation criteria
obtained by SCM are close to be necessary. To this end we will use
the following recent non-oscillation criterion.
\begin{theorem}[\cite{k2}] \label{thm4}
Let (a1)-(a2) hold
and suppose that
there exist $t_0\geq 0$ and positive numbers $x_1,x_2$ such that
for $t\geq t_0$ we have
\begin{equation}
\label{56}
\begin{array}{c}
\ln x_1-x_1\int_{r_1(t)}^t a_1(s)ds-x_2\int_{r_1(t)}^t a_2(s)ds\geq 0 \\[3pt]
\ln x_2-x_1\int_{r_2(t)}^t a_1(s)ds-x_2\int_{r_2(t)}^t a_2(s)ds\geq 0
\end{array}
\end{equation}
Then (\ref{15}) has a non-oscillatory solution.
\end{theorem}
Denote
$$
A_{ij}:=\lim_{t\to\infty}\sup\int_{r_i(t)}^t a_j(s)ds,\quad i,j=1,2.
$$
\begin{corollary} \label{cor4.1}
Let $A_{ij} 0\\
\ln x_2-A_{21}x_1-A_{22}x_2> 0
\end{array}
\end{equation}
Then (\ref{15}) has a non-oscillatory solution.
\end{corollary}
\begin{corollary} \label{cor4.2}
Suppose
\begin{equation} \label{64}
\lim_{t\to\infty}\int_{r_2(t)}^t a_1(s)ds=
\lim_{t\to\infty}\int_{r_2(t)}^t a_2(s)ds=0
\end{equation}
and denote, as before,
$$
A_{11}:=\lim_{t\to\infty}\sup\int_{r_1(t)}^t a_1(s)ds,\quad
A_{12}:=\lim_{t\to\infty}\sup\int_{r_1(t)}^t
a_2(s)ds.
$$
If
\begin{equation}
\label{65}
A_{11}\exp\{A_{12}\} 0\\
\ln x_2> 0
\end{array}\end{equation}
Put $x_1:=\frac{1}{A_{11}}$ (in case $A_{11}>0$). Then (\ref{66}) turns into
$A_{12}0$.
In case $A_{12}=0$ Condition (\ref{65}) turns into $A_{11}1$ will be a solution of (\ref{66}).
The solvability of (\ref{66}) in case $ A_{11}=0$ is an obvious fact.
As in Example 3.1, consider (\ref{46}),
where $a_1, a_2 >0$, $r(t)\leq t$, $\lim_{t\to\infty}[t-r(t)]=0$.
We have $A_{11}=a_1$, $A_{12}=a_2$, $A_{21}=A_{22}=0$.
Then the condition
$a_1e^{a_2}\frac{1}{e}
$
implies that all solutions of (\ref{46}) are oscillatory.
As in Example 3.2, consider (\ref{49}),
where $\mu>1$, $\tau>0$, $a_1, a_2>0$.
We have
$A_{11}=a_1\ln \mu$, $A_{12}=a_2\ln \mu$, $A_{21}=A_{22}=0$.
Hence if the condition
$a_1\mu^{a_2}\frac{1}{e\ln\mu}$ holds then all solutions of
(\ref{49}) are oscillatory.
\section{Discussion}
In this section we compare the results obtained in this paper
with some known results.
\paragraph{Statement 5.1 \cite{a1}} Let $r_i(t):=t-\tau_i$, $\tau_i>0$,
and $\lim_{t\to\infty}\inf \int_t^{t+\tau_i}a_i(s) ds >0$.
Suppose that at least one of the following three conditions holds:
\begin{enumerate}
\item
$p_{ij}^{\ast}= \lim_{t\to\infty}\inf \int_{t-\tau_i}^t a_j(s) ds
>1/e$, for some $i,j$;
\item
$\big[ \prod_{i=1}^n \sum_{j=1}^np_{ij}^{\ast}\big]^{1/n} >1/e$;
\item
$\sum_{i=1}^np_{ij}^{\ast}+2\sum_{in/2$, for some $j$.
\end{enumerate}
Then all solutions of (\ref{2}) are oscillatory.
\paragraph{Statement 5.2 \cite{h1}} Let $a_i(t)>0$, $0\frac{1}{e}.
$$
Then all solutions of (\ref{2}) are oscillatory.
\paragraph{Statement 5.3 \cite{g1}} Assume that
there exist indices $i_l\in\{1,\dots,n\}$ such that
\begin{equation}
\label{74}
\lim_{t\to\infty}\inf (t-r_{i_l}(t))>0,\quad
\lim_{t\to\infty}\inf \sum_{i=1}^n a_{i_l}(t)>0.
\end{equation}
If
$$
\lim_{t\to\infty}\inf \Big[\inf_{\lambda >0} \Big\{
\frac{1}{\lambda}\sum_{i=1}^n a_i(t)\exp\{\lambda (t-r_i(t))\}\Big\}
\Big]>1,
$$
then all solutions of (\ref{2}) are oscillatory.
\begin{corollary}
Assume that (\ref{74}) holds and
\begin{equation}
\label{75}
\lim_{t\to\infty}\inf \Big\{\big[\prod_{i=1}^n a_i(t)\big]^{1/n}
\big[\sum_{i=1}^n(t-r_i(t))\big]\Big\}>\frac{1}{e}.
\end{equation}
Then all solutions of (\ref{2}) are oscillatory.
\paragraph{Statement 5.4 \cite{y1}}
Suppose that for sufficiently large $T$ and for some $\lambda>0$,
$$
-\lambda +\sup_{t\geq T} \sum_{i=1}^n a_i(t)\exp\{\lambda (t-r_i(t))\}\leq 0.
$$
Then there exists a non-oscillatory solution of (\ref{2}).
\end{corollary}
\paragraph{Statement 5.5 \cite{p1}}
Suppose there exist a nonempty set $I\subset \{1,\dots, n\}$ and
constants $\tau_0, \tau_1$, $\tau_0>\tau_1>0$, such that
$$
t-r_i(t)\geq \tau_0,\quad i\in I,\quad \lim_{t\to\infty}\inf \int_t^{t+\tau_1}
\sum_{t\in I} a_i(s) ds >0,
$$
\begin{equation}
\label{76}
\lim_{t\to\infty}\sup \Big\{ \max_{k}
\int_{r_k(t)}^t \sum_{k=1}^n a_k(s)ds\Big\}0$ and some $T>0$,
$$
-\lambda+\inf_{t\geq T} \frac{\sum_{k=1}^n a_k(t)\exp\left\{\lambda
\int_{r_k(t)}^t \sum_{i=1}^n a_i(s)ds\right\}}{\sum_{k=1}^n a_k(t)}>0.
$$
Then all solutions of (\ref{2}) are oscillatory.
\begin{corollary}
Suppose (\ref{76}) hold and
$$
\lim_{t\to\infty}\inf \frac{\sum_{k=1}^n a_k(t) \int_{r_k(t)}^t \sum_{i=1}^n
a_i(s)ds}{\sum_{k=1}^n a_k(t)}>\frac{1}{e}.
$$
Then all solutions of (\ref{2}) are oscillatory.
\end{corollary}
\paragraph{Statement 5.6 \cite{p1}}
Suppose there exist $\lambda>0$ and a sufficiently large $T$ such that
$$
-\lambda+\sup_{t\geq T} \frac{\sum_{k=1}^n a_k(t)\exp\left\{\lambda
\int_{r_k(t)}^t \sum_{i=1}^n a_i(s)ds\right\}}{\sum_{k=1}^n a_k(t)}\leq 0.
$$
Then there exists a non-oscillatory solution of (\ref{2}).
\paragraph{Statement 5.7 \cite{p2}}
Suppose that $t-r_i(t)=\tau_i>0$ and for every $\lambda>0$ and
sufficiently large $T$,
$$
-\lambda+\inf_{t\geq T}\min_{j=1,\dots,n} \sum_{k=1}^n p_{jk}(t)e^{\lambda\tau_k}>0,
$$
where
$p_{jk}(t)=\frac{1}{\tau_j}\int_{t-\tau_j}^t a_k(s)ds$.
Then all solutions of (\ref{2}) are oscillatory.
\begin{corollary}
If
$$
\lim_{t\to\infty}\inf\min_{j=1,\dots,n} \sum_{k=1}^n p_{jk}(t)>\frac{1}{e}
$$
then all solutions of (\ref{2}) are oscillatory.
\end{corollary}
\paragraph{Statement 5.8 \cite{p2}}
Suppose that $t-r_i(t)=\tau_i>0$ and there exist $\lambda>0$ and a sufficiently large $T$
such that
$$
-\lambda+\sup_{t\geq T}\max_{j=1,\dots,n} \sum_{k=1}^n p_{jk}(t)e^{\lambda\tau_k}\leq 0.
$$
Then there exists a non-oscillatory solution of (\ref{2}).
Denote $\tau_k(t)=t-r_k(t)$.
\paragraph{Statement 5.9 \cite{e1}}
Suppose (\ref{74}) holds and for every $\lambda>0$ and $i=1,\dots,n$,
$$
\lim_{t\to\infty}\inf\frac{1}{\lambda\tau_i(t)}\sum_{k=1}^n
\int_t^{t+\tau_k(t)}a_k(s)e^{\lambda\tau_k(t)}ds>1.
$$
Then all solutions of (\ref{2}) are oscillatory.
\begin{corollary}
Suppose (\ref{74}) holds and for every $i=1,\dots,n$,
$$
\lim_{t\to\infty}\inf\frac{1}{\tau_i(t)}\sum_{k=1}^n
\int_t^{t+\tau_k(t)}a_k(s)\tau_k(t)ds>\frac{1}{e}.
$$
Then all solutions of (\ref{2}) are oscillatory.
\end{corollary}
In the recent papers \cite{l2,l3}
Li has generalized some results of Erbe and Kong \cite{e1}.
An interesting oscillation criterion was obtained in \cite{k2}.
Unfortunately, in this paper there is no example illustrating
the strength of this result.
Note also the original paper of Nadareishvili \cite{n1},
where the author does not assume
that $a_k(t)$ are nonnegative functions and $r_k(t)\leq t$.
Most of the results listed above generalized the well-known assertion
on the oscillation of autonomous equations (see Corollary \ref{cor2.3}).
All these results contained
some restrictions on the parameters of the equations.
For example, in \cite{a1,p2}, the authors consider
equations with constant delays,
in \cite{h1} with bounded delays, and in \cite{e1,g1,k2}
under condition (\ref{74}).
These conditions are rather restrictive.
Indeed, because of these restrictions one cannot get oscillation
criteria for equations (\ref{49}), (\ref{51}) and (\ref{52}).
On the contrary, the results of this paper give an almost full description
of the oscillatory properties of these equations.
For example, if $a_1\mu^{a_2} >\frac{1}{e\ln\mu}$,
then all solutions of (\ref{49}) are oscillatory.
If $a_1\mu^{a_2} 0,
\end{equation}
where $a_1,a_2,\tau >0$, $\mu >1$.
For this equation one cannot use the trick applied in Ex.3.3 for the
similar (\ref{51}),
since the condition $B_{11}+B_{22}\neq 0$ in Theorem \ref{thm3}
does not hold for (\ref{77}).
Apply now the Corollary of Statement 5.3.
For $n=2$ Condition (\ref{75}) has the form
$$
\left[a_1(t)a_2(t)\right]^{1/2}\left(\tau_1(t)+\tau_2(t)\right)>\frac{1}{e},
$$
for $t\geq T$, where $T$ is sufficiently large.
Rewrite this condition for (\ref{77}):
\begin{equation}
\label{78}
\left[\frac{a_1a_2}{t^{3/2}}\right]^{1/2}
\left((1-1/\mu)t+\tau\right)>\frac{1}{e}.
\end{equation}
It is obvious that (\ref{78}) holds for $ a_1>0, a_2>0$
and $t$ sufficiently large.
Hence the Corollary of Statement 5.3 implies that for every $a_1>0$ and $a_2>0$
all solutions of (\ref{77}) are oscillatory.
\paragraph{Acknowledgement}
The authors are very grateful to the referee whose valuable remarks
helped us improving this article.
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\noindent\textsc{Leonid Berezansky} (e-mail: brznsky@cs.bgu.ac.il)\\
\textsc{Yury Domshlak} (e-mail: domshlak@cs.bgu.ac.il)\\[3pt]
Department of Mathematics \\
Ben-Gurion University of the Negev \\
Beer-Sheva 84105, Israel
\end{document}